Distributions Histograms A Normal aka Gaussian distribution image

Distributions / Histograms A Normal (aka Gaussian) distribution (image from Mathworld)

http: //www. hccfl. edu/facultyinfo/aambrioso/sta-2023 h. aspx

`Normal’ or Gaussian distributions … • … tend to be everywhere • Given a typical numeric field in a typical dataset, it is common that most values are centred around a particular value (the mean), and the proportion with larger or smaller values tends to tail off.

`Normal’ or Gaussian distributions … • We just saw this – fields 7— 10 were Normal-ish • Heights, weights, times (e. g. for 100 m sprint, for lengths of software projects), measurements (e. g. length of a leaf, waist measurement, coursework marks, level of protein A in a blood sample, …) all tend to be Normally distributed. Why? ?

Sometimes distributions are uniform Uniform distributions. Every possible value tends to be equally likely

This figure is from: http: //mathworld. wolfram. com/Dice. html One die: uniform distribution of possible totals: But look what happens as soon as the value is a sum of things; The more things, the more Gaussian the distribution. Are measurements (etc. ) usually the sum of many factors?

Probability Distributions • If a population (e. g. field of a dataset) is expected to match a standard probability distribution then a wealth of statistical knowledge and results can be brought to bear on its analysis • statistics of a sample provide info about a sample • But if we can assume that our statistic is normally distributed in the population, then our sample statistic provides info about the population

The power of assumptions… You are a random sample of 30 HWU/Riccarton students The mean height of this sample is 1. 685 cm There are 5, 000 students in the population With no more information, what can we say about the mean height of the population of 5, 000 students?

A closer look at the normal distribution This is the ND with mean mu and std sigma

More than just a pretty bell shape Suppose the standard deviation of your sample is 0. 12 Theory tells us that if a population is Normal, the sample std is a fairly good guess at the population std So, the sample STD is a good estimate for the population STD So we can say, for example, that ~95% of the population of 5000 students (4750 students) will be within 0. 24 m of the population mean

But what is the population mean? Mean of our sample was 1. 685 The Standard Error of the Mean is pop std / sqrt(sample size) which we can approximate by: sample std / sqrt(sample size) … in our case this 0. 12/5. 5 = 0. 022 This ‘standard error’ (SE) is actually the standard deviation of the distribution of sample means We can use this it to build a confidence interval for the actual population mean. Basically, we can be 95% sure that the pop mean is within 2 SEs of the sample mean …

The power of assumptions… You are a random sample of 30 HWU/Riccarton students The mean height of this sample is 1. 685 cm There are 5, 000 students in the population With no more information, what can we say about the mean height of the population of 5, 000 students? If we assume the population is normally distributed …. our sample std (0. 12) is a good estimate of the pop std …. . so, means of samples of size 30 will generally have their own std, of 0. 022 (calculated on last slide) … so, we can be 95% confident that the pop mean is between 1. 641 and 1. 729 (2 SEs either side of the sample mean)

This is a good time to mention: Z-normalisation (converting measurements to z-scores) Given any collection of numbers (e. g. the values of a particular field in a dataset) we can work out the mean and the standard deviation. Z-score normalisation means converting the numbers into units of standard deviation.

Simple z-normalisation example values 2. 8 17. 6 4. 1 12. 7 3. 5 11. 8 12. 2 11. 1 15. 8 19. 6

Simple z-normalisation example values 2. 8 17. 6 4. 1 12. 7 3. 5 11. 8 12. 2 11. 1 15. 8 19. 6 Mean: 11. 12 STD: 5. 93

Simple z-normalisation example values Mean subtracted -8. 32 2. 8 17. 6 4. 1 Mean: 11. 12 STD: 5. 93 6. 48 -7. 02 12. 7 1. 58 3. 5 -7. 62 11. 8 0. 68 12. 2 11. 1 subtract mean, so that these are centred around zero 1. 08 -0. 02 15. 8 4. 68 19. 6 8. 48

Simple z-normalisation example values Mean subtracted In Z units -8. 32 -1. 403 6. 48 1. 092 -7. 02 -1. 18 12. 7 1. 58 0. 27 3. 5 -7. 62 -1. 28 11. 8 0. 68 0. 11 2. 8 17. 6 4. 1 12. 2 11. 1 Mean: 11. 12 STD: 5. 93 subtract mean, so that these are centred around zero 1. 08 -0. 02 15. 8 4. 68 19. 6 8. 48 Divide each value by the std; we now see how usual or unusual each value is 0. 18 -0. 003 0. 79 1. 43

Data Mining (and machine learning) DM Lecture 6: Confusion, Correlation and Regression

Today • A bit more basic statistics: Correlation – Undestanding whether two fields of the data are related – Can you predict one from the other? – Or is there some underlying cause that affects both? • Basic Regression: – Very, very often used – Given that there is a correlation between A and B (e. g. hours of study and performance in exams; height and weight ; radon levels and cancer, etc … this is used to predict B from A. – Linear Regression (predict value of B from value of A) – But be careful …

Correlation Are these two things correlated? Phone use (hrs) Life expectancy 1 84 2 78 3 91 4 79 5 69 6 80 7 76 8 80 9 75 10 70

Correlation

What about these (web credit) David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail. com These slides and related resources: http: //www. macs. hw. ac. uk/~dwcorne/Teaching/dmml. html

What about these (web credit)

Correlation Measures It is easy to calculate a number that tells you how well two things are correlated. The most common is “Pearson’s r” The r measure is: r = 1 for perfectly positively correlated data (as A increases, B increases, and the line exactly fits the points) r = -1 for perfectly negative correlation (as A increases, B decreases, and the line exactly fits the points)

Correlation Measures r = 0 No correlation – there seems to me not the slightest hint of any relationship between A and B More general and usual values of r: if r >= 0. 9 (r <= -0. 9) -- a `strong’ correlation else if r >= 0. 65 (r <= -0. 65) -- a moderate correlation else if r >= 0. 2 (r <= -0. 2) -- a weak correlation,

Calculating r You will remember the Sample standard deviation, when you have a sample of n different values whose mean is Sample Std is square root of

Calculating r If we have pairs of (x, y) values, Pearson’s r is: Interpretation of this should be obvious (? )

Correlation (Pearson’s R) and covariance As we just saw, this is Pearson’s R

Correlation (Pearson’s R) and covariance And this is the covariance between x and y often called cov(x, y)

r = 0. 772 cov(x, y) = 6. 15 12 10 R = − 0. 556 cov(x, y) = − 4. 35 10 9 8 7 8 6 5 6 4 4 3 2 2 1 0 0 0 2 4 6 8 10 12 r = 0. 772 cov(x, y) = 61. 5 120 100 0 2 4 0 20 40 10 6 8 10 60 80 100 12 R = − 0. 556 cov(x, y) = − 43. 5 9 8 7 80 6 60 5 4 40 3 2 20 1 0 0 2 4 6 8 10 12 0 David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail. com These slides and related resources: http: //www. macs. hw. ac. uk/~dwcorne/Teaching/dmml. html 120

So, we calculate r, we think there is a correlation, what next?

We find the ‘best’ line through the data

We can now make predictions! We have just done ‘regression’

So, how do we calculate the best line?

Best line means … • We want the “closest” line, which minimizes the error. I. e. it minimizes the degree to which the actual points stray from the line. For any given point, its distance from the line is called a residual. • We assume that there really is a linear relationship ( e. g. for a top long distance runner, time = miles x 5 minutes), but we expect that there is also random `noise’ which moves points away from the line (e. g. different weather conditions, different physical form, when collecting the different data points). We want this noise (deviations from line) to be as small as possible. • We also want the noise to be unrelated to A (as in predicting B from A) – in other words, we want the correlation between A and the noise to be 0. David Corne, and Nick Taylor, Heriot-Watt University - dwcorne@gmail. com These slides and related resources: http: //www. macs. hw. ac. uk/~dwcorne/Teaching/dmml. html

Noise/Residuals: the residuals

• We want the line, which minimizes the sum of the (squared) residuals. • And we want the residuals themselves to have zero correlation with the variables

When there is only one x, it is called univariate regression, and closely related to the correlation between x and y. In this case the mathematics turns out to be equivalent to simply working out the correlation, plus a bit more

Calculating the regression line in univariate regression • First, recall that any line through (x, y) points can be represented by: where m is the gradient, and c is where it crosses the y-axis at x=0

Calculating the regression line To get m, we can work out Pearson’s r, and then we calculate: to get c, we just do this: Job done

Now we can: • Try to gain some insight from m – e. g. “since m = − 4, this suggests that every extra hour of watching TV leads to a reduction in IQ of 4 points” “since m = 0. 05, it seems that an extra hour per day of mobile phone use leads to a 5% increase in likelihood to develop brain tumours”.

BUT • Be careful -- • It is easy to calculate the regression line, but always remember what the r value actually is, since this gives an idea of how accurate is the assumption that the relationship is really linear. • So, the regression line might suggest: “… extra hour per day of mobile phone use leads to a 5% increase in likelihood to develop brain tumours” but if r ~ 0. 3 (say) we can’t say we are very confident about this. Either not enough data, or the relationship is different from linear

And … what about a dataset with more than one non-class field?

The general solution to multiple regression Suppose you have a numeric dataset, e. g. 3. 1 3. 2 4. 5 4. 1 … 2. 1 1. 8 5. 1 4. 1 3. 9 2. 8 2. 4 … 2. 0 1. 5 3. 2 … 6. 0 7. 4 8. 0 8. 2 … 7. 1 6. 2 9. 5

The general solution to multiple regression Suppose you have a numeric dataset, e. g. X 3. 1 3. 2 4. 5 4. 1 … 2. 1 1. 8 5. 1 4. 1 3. 9 2. 8 2. 4 … 2. 0 1. 5 3. 2 … 6. 0 7. 4 8. 0 8. 2 … 7. 1 6. 2 9. 5 y

The general solution to multiple regression Suppose you have a numeric dataset, e. g. X y 3. 1 3. 2 4. 5 4. 1 … 2. 1 1. 8 5. 1 4. 1 3. 9 2. 8 2. 4 … 2. 0 1. 5 3. 2 … 6. 0 7. 4 8. 0 8. 2 … 7. 1 6. 2 9. 5 A linear classifier or regressor is: yn = β 1 xn 1 + β 2 xn 2 + … βmxnm - So, how do you get the β values?

The general solution to multiple regression By straightforward linear algebra. Treat the data as matrices and vectors …

The general solution to multiple regression By straightforward linear algebra. Treat the data as matrices and vectors … And the solution is:

And that’s general multivariate linear regression. If the data are all numbers, you can get a prediction for your ‘target’ field by deriving the beta values with linear algebra. Note that this is regression, meaning that you predict an actual real number value, such as wind-speed, IQ, weight, etc…, rather than classification, where you predict a class value, such as ‘high’, ‘low’, ‘clever’, … But … easy to turn this into classification … how?

Pros and Cons Arguably, there’s no ‘learning’ involved, since the beta values are obtained by direct mathematical calculations on the data It’s pretty fast to get the beta values too. However In many, many cases, the ‘linear’ assumption is a poor one – the predictions simply will not be as good as, for example, decision trees, neural networks, k-nearest-neighbour, etc … The maths/implementation involves getting the inverses of large matrices. This sometimes explodes.

Prediction Commonly, we can of course use the regression line for prediction: So, predicted life expectancy for 3 hrs per day is: 82 - fine

BUT If correlation is not strong, the regression line may be meaningless What would we predict for 8 hrs if we did regression based only on the first cluster?

… or, outliers may be distorting the picture, … or, a straight line is simply not an appropriate model for the data r is 0. 816 y = 3. 00 + 0. 500 x Anscombe, Francis J. (1973) Graphs in statistical analysis. American Statistician, 27, 17– 21.

Correlation and regression are used everywhere in science and industry.

But see here … http: //www. tylervigen. com/spurious-correlations